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Inferring in absence of background

The solution of Eq.(4) depends, at least in principle, on the assumption on the prior . Taking a flat prior between 0 and 1, that models our indifference on the possible values of before we take into account the result of the experiment in which successes were observed in trials, we get (see e.g. [2]):
 (12)

some examples of which are shown in Fig. 1.
Expected value, mode (the value of for which has the maximum) and variance of this distribution are:
 (13) (14) (15) (16)

Eq. (13) is known as recursive Laplace formula'', or Laplace's rule of succession''. Not that there is no magic if the formula gives a sensible result even for the extreme cases and for all values of (even if !). It is just a consequence of the prior: in absence of new information, we get out what we put in!

From Fig. 1 we can see that for large numbers (and with far from 0 and from ) tends to a Gaussian. This is just the reflex of the limit to Gaussian of the binomial. In this large numbers limit and .

Subsections

Next: Meaning and role of Up: Inferring the success parameter Previous: The binomial distribution and
Giulio D'Agostini 2004-12-13